1. Field
The present disclosure relates to lithographic masks and more specifically to a method and system to perform an equi-brightness optimization.
2. Description of Related Art
Typically, a mask optimization problem for a given source, such as an optical source, in a projection nanolithography imaging system may be written as a quadratic problem
                                          a            *                    =                      argmin            ⁢                                          ∑                                  v                  =                  1                                                  v                  max                                            ⁢                                                a                  T                                ⁢                                  c                  v                                ⁢                                  A                  v                                ⁢                a                                                    ,                                  ⁢                                            a              T                        ⁢                          A              u                        ⁢            a                    ≥          1                ,                  1          ≤          u          ≤                      u            max                                              (        1        )            
where aεRn is a vector with the amplitude of all of the diffraction orders in a mask, the diffraction orders being transferable through a partially coherent projection nanolithography imaging system. In particular, a* represents an amplitude of diffraction orders in an optimum mask. A mask may be fabricated, without more, with the information in a*.
Matrices Au and Av are specific to the given source and provide an imaging system model for bright points u and dark points v at an image plane. Specifically, aTAia represents an intensity at point i due to a mask having diffraction orders given by a and an imaging system model (having a set source, for example) given by Ai. Positions of bright points 1, 2 . . . umax are decided based upon a pattern for printing. Initially, the bright points 1, 2 . . . umax may be substantially uniformly distributed within bright regions of the pattern for printing. Similarly, positions of dark points 1, 2 . . . vmax are decided based upon the pattern for printing. Initially, the dark points 1, 2 . . . vmax may be substantially uniformly distributed within dark regions of the pattern for printing. Coefficients cv for the dark points act as a weighting factor, often emphasizing the dark points closer to the bright regions.
A detailed expression for Ai depends on an optical source pattern, an intensity and a polarization of the optical source, fundamental properties of the projection nanolithography imaging system and a location of the bright points and dark points which in turn depends on an ideal wafer pattern.